Properties of geometric progression

In this section we will discuss important properties of geometric progressions and geometric series.
Property I : If all the terms of G.P be multiplied or divided by same non-zero constant then the sequence remains in G.P with the same common ratio.
Proof : Let $a_{1}, a_{2},a_{3},...,a_{n}$ be in geometric progression with common ratio 'r'.
$\frac{a_{n+1}}{a_{n}}$ , for all n $\epsilon$ N ------(i)

Let 'k' be any non-zero constant by which we are multiplying each term of G.P.so we get,
$ka_{1}, ka_{2},ka_{3},...,ka_{n}$

So equation (i)⇒ $\frac{ka_{n+1}}{ka_{n}}$ , for all n $\epsilon$ N

which is same as $\frac{a_{n+1}}{a_{n}}$
&ther4; the new sequence formed is also in G.P. with common ratio 'r'.

Property II : The reciprocals of the terms of a given G.P. form a G.P.
Proof : Let $a_{1}, a_{2},a_{3},...,a_{n}$ be in geometric progression with common ratio 'r'.
$\frac{a_{n+1}}{a_{n}}$ , for all n $\epsilon$ N ------(i)

Now the sequence formed by reciprocal will be
$\frac {1}{a_{1}}, \frac {1}{a_{2}},\frac {1}{a_{3}},...,\frac {1}{a_{n}}$

The common ratio = $\frac{\frac{1}{a_{n+1}}}{\frac{1}{a_{n}}} = \frac{a_{n+1}}{a_{n}}=\frac{1}{r}$
So the new sequence is in G.P. with common ratio $\frac{1}{r}$

Property III : If each term of G.P. raised by the same exponent, the resulting sequence also in G.P.
Proof : Let $a_{1}, a_{2},a_{3},...,a_{n}$ be in geometric progression with common ratio 'r'.
$\frac{a_{n+1}}{a_{n}}$ , for all n $\epsilon$ N ------(i)

Let each term of G.P. is raised by exponent 'k' then the sequence becomes
$a_{1}^{k}, a_{2}^{k},a_{3}^{k},...,a_{n}^{k}$

$\frac{a_{n+1}^{k}}{a_{n}^{k}} = \left ( \frac{a_{n+1}}{a_{n}} \right )^{k} = r^{k}$ for all n $\epsilon$ N
∴ $a_{1}^{k}, a_{2}^{k},a_{3}^{k},...,a_{n}^{k}$ is in G.P with common ration $r^{k}$

Properties of geometric progression

Property IV : In a finite G.P. the product of the terms equidistant from the beginning and the end is always same and is equal to the product of first and the last term.
Proof : Let $a_{1}, a_{2},a_{3},...,a_{n}$ be in geometric progression with common ratio 'r'.
kth term from the beginning = $a_{k} = a_{1}r^{k - 1}$
kth term from the end = $a_{n - k + 1} = a_{1}r^{n - k}$
∴ product of kth term from the beginning and kth term from the end
= $a_{1}r^{k - 1} \times a_{1}r^{n - k}$
= $a_{1}^{2} r^{k - 1 + n - k} $
= $a_{1}^{2} r^{n -1} $ for k = 2,3,4,...(n -1)
Hence the product of the terms equidistant from the beginning and the end is always same and is equal to the product of first and the last term.

Property V : Three non-zero numbers a,b,c are in G.P. if and only if $b^{2}$ = ac
Proof : a, b c are in G.P.
∴ common ration = $\frac{b}{a} = \frac{c}{a} $
∴ $b^{2}$ = ac

Property VI : If the terms of a given G.P. are chosen at regular intervals, then the new sequence so formed also forms G.P.

Property VII : If $a_{1}, a_{2},a_{3},...,a_{n}$ be in geometric progression, then $ log a_{1}, log a_{2},log a_{3},...,log a_{n}$ is an arithmetic progression and vice-versa.
11th Grade math

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